A student follows the rule `3 xx 3 + 3 xx 3 = 12, 4 xx 4 + 4 xx 4 = 16`, `5 xx 5 + 5 xx 5 = 20`, then find the value of the expression `12 xx 12 + 12 xx 12` according to this rule.

Options

A. 288
B. 48
C. 144
D. 44

Answer

To solve the problem, we need to follow the pattern established in the examples given in the question.

1. **Understanding the Rule**:
The examples provided are:
- \(3 \times 3 + 3 \times 3 = 12\)
- \(4 \times 4 + 4 \times 4 = 16\)
- \(5 \times 5 + 5 \times 5 = 20\)

From these, we notice that:
- \(3 \times 3 = 9\), and \(9 + 9 = 18\) (but they say it equals 12).
- \(4 \times 4 = 16\), and \(16 + 16 = 32\) (but they say it equals 16).
- \(5 \times 5 = 25\), and \(25 + 25 = 50\) (but they say it equals 20).

It seems that the multiplication is being treated differently. Instead of using the regular multiplication, it appears that the result of \(n \times n\) is halved.

2. **Applying the Rule**:
- For \(n = 3\):
\[ 3 \times 3 = 9 \quad \text{(but treated as 6)} \]
- For \(n = 4\):
\[ 4 \times 4 = 16 \quad \text{(but treated as 8)} \]
- For \(n = 5\):
\[ 5 \times 5 = 25 \quad \text{(but treated as 10)} \]

The pattern suggests that \(n \times n\) is being treated as \( \frac{n \times n}{2} \).

3. **Finding the Value for \(12 \times 12\)**:
Now, we need to find \(12 \times 12 + 12 \times 12\):
- Calculate \(12 \times 12\):
\[ 12 \times 12 = 144 \]
- According to the pattern, we take half of this:
\[ \frac{144}{2} = 72 \]
- Now, we need to add this result to itself:
\[ 72 + 72 = 144 \]

4. **Final Calculation**:
Therefore, the value of the expression \(12 \times 12 + 12 \times 12\) according to the rule is:
\[ 144 \]